In this case one often says that (or ) is a Frobenius functor.
There is no relation to the notion of Frobenius monoidal functor.
K. Morita proved that the extension of scalars functor for a morphism of rings is Frobenius iff the morphism itself is Frobenius in the sense of (Kasch), that is: is finitely generated projective and as -bimodules.
This is in the spirit of the finite-dimensional duality coded e.g. in the notion of Frobenius algebra.
Stefaan Caenepeel, Gigel Militaru, Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Springer Lec. Notes in Math. 1787 (2002) xiv+354 pp, gBooks
F. Kasch, Projektive Frobenius-Erweiterungen, Sitzungsber, Heidelberger Akad. Wiss., Math.- Naturw. Kl. 1960/61 (1961), 89–109.